Solving Equations: A Step-by-Step Guide

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Solving Equations: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of equations, specifically, how to solve a system of equations. Don't worry, it's not as scary as it sounds! We'll break it down step-by-step and make sure you understand the core concepts. We will solve the system of equations:

y=x−3y=x2−5x+6\begin{array}{l} y=x-3 \\ y=x^2-5 x+6 \end{array}

So, grab your pens and paper, and let's get started!

Understanding the Basics: What are Equations and Systems?

Alright, before we jump into the nitty-gritty of solving the system of equations, let's get a handle on what we're actually dealing with. At its heart, an equation is a mathematical statement that asserts the equality of two expressions. Think of it like a balanced scale: whatever's on one side must equal what's on the other. Equations can involve numbers, variables (like x and y), and mathematical operations (+, -, ×, ÷, etc.). The goal is usually to find the value(s) of the variable(s) that make the equation true.

Now, a system of equations is simply a set of two or more equations that we want to solve simultaneously. This means we're looking for the values of the variables that satisfy all the equations in the system. The solution to a system of equations represents the point(s) where the graphs of the equations intersect. In the simplest form, the solution to a linear system of two equations is a single point. A point where the two lines intersect. This point can also be no point if the lines are parallel. This is where we will start today, with our system of equations:

y=x−3y=x2−5x+6\begin{array}{l} y=x-3 \\ y=x^2-5 x+6 \end{array}

In our case, we have two equations with two variables, x and y. Our mission? Find the values of x and y that make both equations true at the same time. These two equations are graphed differently, one is linear and the other is a quadratic. The linear equation will have a line, and the quadratic will have a parabola. We are looking for where those two lines intersect. Let's move onto the method of solving.

The Substitution Method: Your Secret Weapon

There are several ways to tackle a system of equations, but we'll use one of the most common and efficient methods: the substitution method. The core idea behind substitution is to solve one of the equations for one variable and then substitute that expression into the other equation. This reduces the problem to a single equation with a single variable, which we can then solve. Let's break down the steps:

  1. Choose an Equation and Solve for a Variable: Look at your equations. Sometimes, one equation might already have a variable isolated (i.e., x or y is already on one side of the equation). In our case, the first equation, y = x - 3, already has y isolated. If not, you'll need to manipulate one of the equations to isolate a variable. This first step can already give us the answer to one of the variables. The equation says y is equivalent to x - 3. Now that we know that y is equivalent to x - 3, we can substitute that for all the y's in the second equation.

  2. Substitute: Substitute the expression you found in step 1 into the other equation. In our example, we'll substitute x - 3 for y in the second equation: y = x² - 5x + 6. This will give us * (x - 3) = x² - 5x + 6*.

  3. Solve the New Equation: Now you have a single equation with only one variable (x in our case). Solve this equation for the variable. This might involve simplifying, combining like terms, factoring, or using the quadratic formula. Our new equation is * (x - 3) = x² - 5x + 6*, but it's not quite ready to solve. You will need to move all of the elements to one side to get it into quadratic form. After moving everything to the right side, the equation becomes 0 = x² - 6x + 9. This can be factored by finding two numbers that multiply to 9 and add to -6. In this case, those numbers are -3 and -3. This can be written as (x - 3)(x - 3) = 0. This can also be written as (x - 3)² = 0. Now we can solve for x. The answer is x = 3.

  4. Back-Substitute: Once you've found the value of one variable, substitute that value back into either of the original equations (or the equation where you isolated a variable) to solve for the other variable. Now that we know that x = 3, we can plug it into our first equation. Our first equation is y = x - 3. Replacing x, we have y = 3 - 3. So the answer is y = 0.

  5. Write the Solution: Finally, write your solution as an ordered pair (x, y). This is the point where the two equations intersect. So our answer is (3, 0).

Step-by-Step Solution for Our System

Alright, let's walk through the substitution method step-by-step to solve our system of equations:

y=x−3y=x2−5x+6\begin{array}{l} y=x-3 \\ y=x^2-5 x+6 \end{array}

  1. We're already set! The first equation, y = x - 3, already has y isolated. This is perfect.

  2. Substitute: Substitute x - 3 for y in the second equation:

    x - 3 = x² - 5x + 6

  3. Solve for x: Let's rearrange the equation to get everything on one side. Subtracting x and adding 3 to both sides, we get:

    0 = x² - 6x + 9

    This is a quadratic equation. We can factor it:

    0 = (x - 3)(x - 3)

    This simplifies to:

    0 = (x - 3)²

    Taking the square root of both sides gives us:

    x - 3 = 0

    Therefore, x = 3

  4. Back-Substitute: Now, substitute x = 3 back into the first equation to solve for y:

    y = 3 - 3

    y = 0

  5. Write the Solution: The solution to the system is (3, 0). This means the graphs of the two equations intersect at the point (3, 0).

Visualizing the Solution: Graphs and Intersection

It's always a good idea to visualize what's happening. The solution to a system of equations represents the point(s) where the graphs of the equations intersect. Let's briefly look at the graphs of our two equations.

  • The first equation, y = x - 3, is a straight line. It has a slope of 1 and a y-intercept of -3. Graphically, it's a diagonal line going upwards from left to right.
  • The second equation, y = x² - 5x + 6, is a parabola. It opens upwards. We already know it intersects the line at (3,0). It's a U-shaped curve.

When you graph these two equations, you'll see that they indeed intersect at the point (3, 0). This intersection point is the solution to our system of equations. In the event you have more than one intersection, the intersection points are the solution. Not all systems will have an intersection, but this system does.

Visualizing the solution reinforces the concept that solving a system of equations means finding the common point(s) that satisfy all the equations in the system. Graphing can also help you understand if the solution is even possible.

Let's Recap!

Alright, let's sum up what we've covered today:

  • We learned that a system of equations is a set of two or more equations that you solve simultaneously.
  • We used the substitution method to solve the system, which involves isolating a variable in one equation and substituting its equivalent expression into the other equation.
  • We systematically solved our specific system of equations and found the solution to be (3, 0).
  • We visualized the solution by understanding the graphical representation, recognizing that the intersection point of the graphs represents the solution.

Solving systems of equations is a fundamental skill in mathematics, with applications in various fields. Keep practicing, and you'll become a pro in no time! Keep on the lookout for other types of equation solvers. There are many different ways to solve equations. Each of them has different strengths and weaknesses. The substitution method is good for simple systems. But other systems may require a different type of solver. Thanks for joining me today! Happy solving, guys!